Anders Björn (Linköping University)

Characterizations and removability of p-superharmonic functions on metric spaces
Abstract.
Superharmonic functions are used as a tool to study harmonic functions. They are, e.g., used to define Perron solutions of the Dirichlet (i.e. boundary value) problem for harmonic functions. In connection with nonlinear p-harmonic functions on weighted R^n, p-superharmonic functions were used in the monograph Heinonen-Kilpeläinen-Martio [1]. An important feature of the definition is that the bounded p-superharmonic functions are exactly the lower semicontinuously regularized supersolutions (there are also unbounded p-superharmonic functions which are not supersolutions).
In metric spaces, Kinnunen-Martio [2], realized that the definition from [1] was not so well suited. They proposed a different definition, which in the (weighted) Euclidean case is equivalent to the definition in [1]. In connection with quasiminimizers, they gave a third definition in [3], and showed that a p-superharmonic function according to the third definition is also p-superharmonic according to the second definition. It turns out that the second and third definitions are equivalent and always imply the first definition, whereas the first definition implies the other two if there are enough regular domains. In this talk I will discuss this and give several other characterizations of superharmonicity in metric spaces. If time permits, I will also discuss removable singularities for p-harmonic and p-superharmonic functions. Relatively closed sets of capacity zero are always removable, but I will discuss new counterexamples of sets of positive capacity (and even positive measure) removable for p-harmonic functions.
References
[1] Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993.
[2] Kinnunen, J. and Martio, O., Nonlinear potential theory on metric spaces, Illinois Math. J. 46 (2002), 857-883.
[3] Kinnunen, J. and Martio, O., Potential theory of quasiminimizers, Ann. Acad. Sci. Fenn. Math. 28 (2003), 459-490.

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